A simplified combined displacement and personalized ventilation model.
The displacement ventilation (DV) system has become popular over recent years for its improved indoor air quality (IAQ) and low energy consumption (Jiang et al. 1992; Yuan et al. 2001). The DV system turns the supply air upside down by allowing the supply fresh air to enter the space through side wall grills located at low levels near the floor, therefore dividing the space into two areas: lower occupied fresh air zone and an upper contaminant zone above the breathing level of occupants (Mossolly et al. 2008). In the DV system, the fresh air is supplied to the occupied zone at a very low velocity of less than 0.2 m/s (39.37 ft/min) and temperatures greater than 18[degrees]C (64.4[degrees]F) near the floor level. The air motion of a DV system is mainly triggered by buoyancy forces, and as the air gets heated, it rises toward the exhaust outlet located at a high level picking the contaminant in its way from the occupied zone. The high air supply temperature compared to the conventional overhead supply air temperature of 12[degrees]C to 14[degrees]C (53.6[degrees]F to 57.2[degrees]F) limits the applicability of the DV system to low cooling loads of less than 40 W/[m.sup.2] (12.7 Btu/h.[ft.sup.2]). For higher cooling capacities, a combined chilled ceiling and displacement ventilation (CC/DV) is recommended. Behne's (1999) design diagram sets the cooling load limit of DV when combined with a chilled ceiling to 100 W/[m.sup.2] (31.7 Btu/h.[ft.sup.2]) of floor area. The chilled ceiling carries a portion of the sensible cooling load, and the DV system carries the rest of the sensible load in addition to the latent cooling loads. Although, the combined CC/DV system is capable of extending the cooling load range of the DV system, the equipment cost and operational complexity of the integrated system would substantially increase.
To make use of the good IAQ of the DV without having to combine it with a chilled ceiling for higher cooling loads, a personalized ventilator (PV) can be combined with the DV system. The PV system can successfully provide the occupant with the privilege of controlling his/her microclimate, temperature, and quality by making use of air modules installed in reachable places near the occupant (desk, furniture) or directly mounted in the floor to provide localized ventilation and comfort to occupants in their work space in offices. The PV system is energy efficient because by directing the cool air to the upper sensitive region of the human body, the overall comfort of the occupant will be improved without the need for having homogenous cool air temperature in the occupant microclimate. Task ventilators are usually designed with an underfloor air distribution system where it could easily direct the conditioned air from the underfloor through concealed ducting systems in the furniture to locations close to the occupant. In DV systems, the implementation of personalized task ventilators might not be feasible because it might require extending air ducting system for occupants located at an appreciable distance from the DV supply grill. In a recent study, Halvonova et al. (2010) explored the idea of using ductless PVs in an office air-conditioned by a DV system by utilizing the lower clean air of the DV system. They reported that supplying cool and clean air at the breathing level of the occupants while maintaining an elevated room temperature may improve the occupants' thermal comfort, the air quality they perceive, as well as the inhaled air quality compared to a standalone operating DV system (Halvonova et al. 2010). In addition, they stated that the use of ductless personalized ventilation in conjunction with DV systems does not affect the vertical temperature gradient and the velocity distribution inside the room (Halvonova et al. 2010). Therefore, the idea of integrating a DV and PV system is viable, and improved energy savings and IAQ can result from such a system.
Assessing the performance and energy savings that could result from such an integrated system can be conducted by experimentation, computational fluid dynamics (CFD) simulations, and modeling. Experimentation and human comfort testing is costly and quite limited to the experimental setup, whereas CFD simulations require expensive computer time. The research on mathematical modeling was restricted to DV systems (Mundt 1992; Carrilho da Graca 2003) and to the integrated CC/DV system (Ayoub et al. 2006). However, little is known about the effect of coupling the task ventilation modules with the DV systems on the space temperature field and plume flow rates. This article presents a model for the integrated DV/PV system derived from fundamental equations of thermal transport from plumes induced by heat sources in the space. The model will predict the temperature distribution inside and outside the plume heat source and the plume flow rates. The DV/PV system will be integrated with a bio-heat model to assess the resulting comfort level and the energy savings that could result from using such a system.
Modeling of a PV-DV combined system
To assess the air vertical temperature gradient and the resulting energy consumption for a PV system in conjunction with a DV system, a mathematical model that can simulate the transient air temperature and flow rates of the thermal plumes and of the surrounding air was developed. Two air volumes are identified (Figure 1) rising thermal plumes (heat source and walls) and the surrounding air. The space wall surfaces are assumed to be isothermal (Carrilho da Graca 2003; Mundt 1992; Linden et al. 1990). In addition, the air movement inside the space is restricted to the rising thermal plumes. Any other movements or possible temperature gradients are neglected, and thus, the space air is divided into horizontal air layers. In each layer, the temperature inside the rising plumes is different than the homogenous surrounding air temperature zone and from the wall plume air temperature. The airflow of the PV could disturb the buoyant plume airflow, and it might create some mixing with the surrounding air zone, especially if the PV air is injected at high air velocity or if the resulting mix temperature between the thermal plumes and the surrounding air is lower than the surrounding air temperature forcing the rising air plume to descend. Such conditions could result in the destruction of the thermal plume rising profiles and would make the use of any thermal plume model unsuitable for predicting the physical and thermal parameters of the integrated DV/PV system airflow. So to be able to consider the rising plumes throughout the different air layers, the PV airflow velocity is assumed to be low enough when reaching the thermal plume so it will not destroy its rising profile, and the resulting temperature mix should be also higher than the surrounding air temperature. The effect of the PV airflow will be only on the plume air temperature and flow rate. Under these assumptions, the mathematical model was developed and coupled with a thermal space model as described below.
[FIGURE 1 OMITTED]
The thermal space model
A thermal space model was developed to simulate the indoor environmental conditions of a space for given outdoor weather data. The model takes as input the outdoor air conditions, wind velocity, and solar radiation. It computes the hourly external convection coefficients and uses them to compute the internal and external wall temperatures through conduction, convection, and radiation heat transfer. The wall temperatures generated by the thermal space model are useful to calculate the wall thermal plume flow rates and the internal convective coefficients, which will be used in the energy balance equation defined in the next section. The internal vertical walls convection coefficients are calculated using the equation
[h.sub.ci] = 1.823[DELTA][T.sup.0.291]/[L.sup.0.1], (1)
while the adopted correlation for the floor or ceiling is that of Min et al. (1956) and is given by
[h.sub.cv,f] = 2.13 [([DELTA]T).sup.0.31]. (1a)
DV and personal ventilation model
Once the internal walls temperatures are calculated, they serve as input for the DV and PV model to simulate the internal air temperature. The air movement is assumed to be unidirectional through the vertical direction. Thus, the space is divided into seven horizontal layers. Each layer is characterized by three homogenous air temperatures: [T.sub.wp], air temperature inside the wall plume; [T.sub.sp], air temperature inside the hot body thermal plume; and [T.sub.a], a surrounding air temperature (see Figure 1). The first three levels are designed according to the human body segments sizes, as shown in Figure 1. The last four levels are divided equally for the remaining distance. The human body is considered to be in the seated position, and it is represented by a vertical cylinder of diameter D = 0.47 m (1.54 ft) and height H = 1.1 m (3.61 ft) so that the total cylinder area would be equal to the average human body surface area of 1.8 [m.sup.2] (19.37 [ft.sup.2]).
The axisymmetric plume flow rates that result from the occupants will be calculated using the properties of an equivalent thermal plume generated by a virtual point source. The model developed by Mundt (1992) will be adopted to calculate the temperature inside the rising human body thermal plume, and the asymmetric wall plumes characteristics (flow rate and temperature) will be calculated using the wall-air temperature difference (Eckert and Jackson 1950).
According to Mundt (1992), the profile of the vertical velocity for the heat source plume is assumed to be similar at all heights and can be represented by a Gaussian profile with U0 as the velocity at the plume centerline:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)
The flow rate of a plume generated by a heat source of flux [P.sub.k] in a space with temperature gradient ([partial derivative][theta]/[partial derivative]z) outside the plumes can be calculated by the following steps (Mundt 1992):
[q.sub.s], = 2.38 x [10.sup.-3][rho][P.sup.3/4.sub.k][([partial derivative][theta]/[partial derivative]z).sup.- 5/8][m.sub.1], (3)
where [m.sub.1] and [z.sub.1] are nondimensional terms defined by
[m.sub.1] = 0.004 + 0.039[z.sub.1] + 0.38[z.sup.2.sub.1] - 0.062[z.sup.3.sub.1], (4)
[z.sub.1] = 2.86(z + [z.sub.v])[([partial derivative][theta]/[partial derivative]z).sup.3/8]/[P.sup.1/4.sub.k], (5)
where z and [z.sub.v] represent the plume height and the virtual point source height, respectively.
Eckert and Jackson (1950) proposed the following velocity profile for the wall plumes having a maximal velocity [U.sub.max] and a boundary layer of thickness [delta],
U = 1.87[U.sub.max][(y/[delta]).sup.1/7][(1 - y/[delta]).sup.4], (6)
whereas the wall plumes flow rate is given by Jaluria (1980) and Etheridge and Sandberg (1996) as
[q.sub.w] = 2.87 x [10.sup.-3][rho][([DELTA][[theta].sub.w]).sup.1/4][z.sup.3/4] Y, (7)
where Y is the wall width, and [DELTA][[theta].sub.w] is the wall-air temperature difference.
Similar to the plume velocity profile, the rising plume-air temperature difference [DELTA][[theta].sub.sp] between the plume air temperature and the surrounding ambient air temperature for the heat source plume can be represented by a Gaussian profile:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)
In an environment with a vertical temperature gradient, the plume air temperature difference is given in terms of the nondimensionalplume air temperature difference [DELTA][[theta].sub.sp,01] (Mundt, 1992) as
[DELTA][[theta].sub.sp,0] = 0.819[[alpha].sup.-1/2.sub.G][F.sup.- 1/4.sub.0][G.sup.5/8]/[[beta].sub.g] [DELTA][[theta].sub.sp,01] (9)
where [F.sub.0] and G, respectively, are defined as [F.sub.0] = [[beta]g][P.sub.k]/[rho][C.sub.p] and G = [[beta]g]([partial derivative][theta]/[partial derivative]z). This would result in an expression of [DELTA][[theta].sub.sp,0] as
[DELTA][[theta].sub.sp,0] = 0.705[P.sup.1/4.sub.k][([partial derivative][theta]/[partial derivative]z).sup.-5/8] [DELTA][[theta].sub.sp,01]. (10)
However, [DELTA][[theta].sub.sp,01], the nondimensional plume-air temperature difference, can be written in terms of the nondimensional variables [f.sub.1] and [m.sub.1] as
[DELTA][[theta].sub.sp,01] = [f.sub.1]/[m.sub.1]. (11)
From Equation 4 and from the relation [partial derivative][f.sub.1]/[partial derivative][z.sub.1] = -[m.sub.1], the nondimensional plume-air temperature difference can be expressed as function of [z.sub.1] alone and becomes
= 1 - 0.004[z.sub.1] - 0.0195[z.sup.2.sub.1] - 0.1266[z.sup.3.sub.1] + 0.0155[z.sup.4.sub.1]/ 0.004 + 0.039[z.sub.1] + 0.38[z.sup.2.sub.1] - 0.062[z.sup.sub.1] (12)
This procedure will make Equation 10 calculable at every height.
[FIGURE 2 OMITTED]
For the wall plumes, the plume-air temperature difference is function of the boundary layer thickness [delta], and of y, the distance from the wall (Eckert and Jackson 1950):
[DELTA][[theta].sub.wp] = [DELTA][[theta].sub.w]([1 - (y/[delta]).sup.1/7), (13)
where [DELTA][[theta].sub.w] is the wall-air temperature difference.
Plumes heat flux
The space is divided into seven horizontal layers with adjacent wall segments, and k corresponds to the air layer index. Each layer interacts with the adjacent side wall segments. The DV air supply enters from the side wall into the floor air layer (k = 1) at flow rate [M.sub.sup] and temperature [T.sub.sup] and the exhaust air leaves from the side wall of the upper air layer at [T.sub.a] (7). The flow rates resulting from walls and heat sources are denoted by [q.sub.w](k) and [q.sub.s](k), respectively. The net circulated mass [M.sub.cir] at each boundary (interface surface between two adjacent air layers k and k + 1) will be calculated (see Figure 2). The enclosure mass balance equation is given by
[M.sub.cir](k) = [M.sub.s] - [summation][q.sub.s](k) - [summation][q.sub.w](k). (14)
To solve the energy balance equations of the enclosure air layers, the values of the airflow rates passing from one layer to the other due to the generated plumes from the heat sources and the walls are calculated from the plume equation for the sources and using the wall plume model tested in this work.
Having found the plume-air temperature difference as a function of the height and the heat source flux, the principle of conservation of energy for each layer can be expressed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This energy balance equation is applied to each air layer k representing the control volume. The [m.sub.i] and [h.sub.i] terms represent the flow rates and enthalpies of the airflows entering layer k (including the wall plumes, heat source plumes, and the circulated mass). Similarly, the [[??].sub.e] and [h.sub.e] represent the flow rates and enthalpies exiting the air layer. [??] represents the external and internal heat sources acting on the control volume. The term [phi] represents the internal heat sources that are the human body segments present at a certain layer and generating a heat flux. The "convective heat transfer" term represents the convective heat flux transferred from the walls area to the control volume through convection. Also, the mass flow rate of the circulated mass could be easily calculated from Equation 14 and its enthalpy easily evaluated since the circulated mass temperature is homogeneous. However, the other terms on the left-hand side of Equation 15 involving the enthalpy of the moving fluid for the thermal plumes are space dependent and follow the Gaussian temperature and flow rate profiles. So the aim of the following is to calculate the enthalpy rate of the thermal plumes defined as [??] = [??]h using infinitesimal calculation.
Assuming an infinitesimal mass flow rate denoted dm, the elementary amount of heat transferred with this flow rate having a certain enthalpy h is given by
d[??] = [C.sub.p]Td[??]. (16)
Since the velocity has a Gaussian profile, and the flow rate is directly proportional to the velocity and to the elementary cross-sectional area dA, the elementary flow rate dm could be expressed as d[??] = [rho]U d A. Since the density variations at a given height are neglected comparing to the velocity variations, the density is assumed to be constant. By grouping the constants and renaming them as [c.sub.1] and [c.sub.2], the elementary flow rates could be written as:
for the source plumes:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
for the wall plumes:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
To find the constants [c.sub.1] and [c.sub.2] in terms of the flow rates [q.sub.s] and [q.sub.w], the fact that the total mass flow rate is equal to the summation of the elementary flow rates is applied and expressed in the following equations: [q.sub.s] = [[integral].sup.R.sub.0] [d[??].sub.s] and [q.sub.w] = [[integral].sup.[delta].sub.0] [d[??].sub.w]. This yields
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
If a dimensional analysis is performed, it is clear that the dimensions of the elementary flow rates are consistent.
Now replacing in Equation 16 and integrating
[[??].sub.s] = [q.sub.s][C.sub.p] (0.684[T.sub.sp,0] + 0.316[T.sub.a]) (20a)
[[??].sub.w] = [q.sub.w][C.sub.p] (0.75[T.sub.a] + 0.25[T.sub.w]) (20b)
The centerline plume temperature [T.sub.sp,0] is calculated from Equation 10, and the wall temperature [T.sub.w] is obtained from the thermal space model, which leaves Equation 15 in terms of [T.sub.a] only, which will be solved for different layers.
Integration of the P V module
For the air layer at the PV module nozzle, the fresh air jet will penetrate the rising thermal plume at a low velocity without disturbing it. The cool air from the PV module will then merge with the pre-existing plume and will reduce its temperature. The new plume flow rates and temperatures will be calculated by adding the flow rates and performing an energy balance on the plumes control volume, respectively. Admitting that, the energy balance equation for the air layer including the PV module is of the following form:
[[??]'.sub.s] = [[??].sub.s] + [[??].sub.PV], (21)
with [[??].sub.PV] = [q.sub.PV][C.sub.p][T.sub.PV] and [[??]'.sub.s] the enthalpy rate of the mixture.
The enthalpy rate of the rising plume merged with the PV flow becomes
[[??]'.sub.s] = [q.sub.s][C.sub.p] (O.684[T.sub.sp,0] + 0.316[T.sub.a]) + [q.sub.PV][C.sub.p][T.sub.PV] (22)
Using Equation 22, the new centerline plume temperature associated with the layer including the personalized ventilation module becomes
[T'.sub.sp,0], [q.sub.s] (0.684[T.sub.sp,0] + 0.316[T.sub.a]) + [q.sub.PV][T.sub.PV] (22) / 0.684([q.sub.s] + [q.sub.PV]) -0.462[T.sub.a]. (23)
The heat exchange between the plume air and the PV air is assumed to be instantaneous. Thus, in order to calculate the heat source plume temperature at the following layers above the PV nozzle, the virtual height z' associated with the plume temperature [T'.sub.sp,0] is calculated at the layer including the PV module] A correction factor equal to the difference between the actual height and the virtual height is introduced as (z - z'). This correction factor is used to obtain the virtual height at the following layers, and the heat source plume temperature is calculated using the newly obtained virtual height z'.
However, this model has some limitations when it comes to the integration of the PV module. Since the heat source plume is assumed to be rising in the vertical direction, the flow rate of the PV should be low enough not to disturb the plume. Moreover, the buoyancy forces responsible for the plumes displacement are triggered by the density difference in the air. So when the PV air temperature is too low compared to the plume temperature, this density difference may vanish and the model would not be able to predict the flow rates and temperatures accurately.
The solution methodology
Since, at each air layer, the energy balance equation is only dependent of the preceding and following layer, the system of equations At = d is of the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
In order to be able to solve this system, the wall temperatures generated by the thermal space model are required. So an iterative method is required, since the DV-PV model and the thermal space model are dependent on each other. As a first step, all the model parameters are initialized and the boundary conditions set (including the outdoor weather conditions, the DV supply flow rate and temperature, and the human body heat flux). The wall temperatures are generated using these initial conditions, and the system could be solved for [T.sub.a].
Since A, the solving matrix of the system, is tridiagonal, then the system could be solved for [T.sub.a] using the tridiagonal matrix algorithm (TDMA) in one dimension. Having solved for the circulating air temperature [T.sub.a], and for the plume temperatures [T.sub.sp] and [T.sub.wp], the thermal properties of the air volumes in the space are defined, and the wall temperatures could be updated accordingly and a new iteration could start. The convergence criterion and the number of iterations are dependent on the tolerance factor of the residuals, which is taken, in this case, to be of the order of [10.sup.-6] (see Figure 3).
[FIGURE 3 OMITTED]
To validate the results generated by the numerical model, a PV module was installed inside a climate chamber equipped with a DV system. The experimental setup, shown in Figure 4, mainly consisted of twin climatic chambers of inner dimensions 2.5 m x 2.75 m x 2.8 m (8.2 ft x 9.02 ft x 9.18 ft). The room walls thermal conductance is 1 [+ or -] 0.1 W/[m.sup.2] x K (0.32 [+ or -] 0.032 Btu/h x [ft.sup.2]- [degrees]F). The grills for supply and return air of the DV system are of a cross-sectional area of 1.91 m (6.27 ft) (width) x 0.32 m (1.05 ft) (height). The supply grill lower edge is at about 0.4 m (1.31 ft) above floor level. The PV module is a circular duct of 0.05 m (2 in.) diameter installed at 0.85 m (2.79 ft) above the floor, and it is equipped with an axial fan at its inlet, which sucks the preconditioned air from the adjacent chamber. The fan is capable of delivering air at flow rates up to 8 L/s (16.95 [ft.sup.3]/min). However, the PV flow rate could be regulated manually by the using the installed damper. The twin climate chambers are installed indoors, and so weather conditions were restricted to an indoor constant temperature of 26[degrees]C (77[degrees]F). The DV system supply air temperature is regulated to the desired value using a proportional-integral-derivative (PID) controller. The model was tested using the experimental cases summarized in Table 1 in the following ways:
* Setting the DV flow rate to 60 L/s (127 [ft.sup.3]/min) and setting the DV air temperature to two different values of 18[degrees]C and 20[degrees]C (64.4[degrees]F and 68[degrees]F) with two different values for the PV module air temperature of 20[degrees]C and 22[degrees]C (68[degrees]F and 71.6[degrees]F) and a PV flow rate of 4 L/s or 7 L/s (8.47 [ft.sup.3]/s or 14.83 [ft.sup.3]/s) and making comparisons between the measured and predicted ambient air column temperatures.
* Using the same parameters defined above to make comparisons between the measured and predicted plume-air temperature difference at the plume's centerline.
[FIGURE 4 OMITTED]
The estimated load on the chamber is 300 W, so a DV flow rate of 60 L/s (127 [ft.sup.3]/min) is chosen to remove this load and typical values for the PV flow rate of 4 L/s and 7 L/s (8.47 [ft.sup.3]/min and 14.83 [ft.sup.3]/min) were chosen. The upper limit of 7 L/s was chosen to be close to the ASHRAE ventilation recommendations for a typical office, and the lower limit of 4 L/s was chosen to maintain a minimum velocity and keep the occupant in thermal comfort (ASHRAE 2009). The experimental test was run on a heated cylinder of diameter 0.47 m (1.54 ft) and height 1.1 m (3.61 ft), as seen in Figure 4. The aim of the experiment was to show the ability of the numerical model to predict the column air temperature and thus the vertical temperature gradient, as well as the thermal plume temperature and the effect of the personalized ventilation.
In order to measure the temperature of the column of air, thermocouples are placed at required locations on wooden rods, as shown in Figure 4. The thermocouples are T-type made by Omega with an accuracy of [+ or -] 0.1[degrees]C (4-0.18[degrees]F). Two other thermocouples are used to measure the dry-bulb and wet-bulb air temperatures at the outlet of the DV supply. The thermocouples are connected to a data acquisition system to collect the data.
Airflow rate measurements
Airflow rates delivered by the PV module and the DV system are estimated using anemometers made by Omega with an accuracy of [+ or -]5%, which are installed in the fans downstream. The anemometers will measure the velocity and the flow rate is calculated using the cross-sectional area. The fans RPM is adjusted to meet the desired flow rate.
Results and discussions
Varying the DV temperature at a desired flow rate while regulating the PV module flow rate and temperature at the values set in Table 1 is an effective method to test the applicability of the model to realistic cases. The layer where the PV air is injected was chosen to be at the upper body part since it has the largest body area fraction and is located near the occupant breathing level for better air quality. Two different typical PV temperatures were chosen to mimic real situations where PV is involved in energy savings.
Figure 5 shows the effect of varying the DV and PV flow rates and temperatures on the surrounding air temperature outside the plume. In Figure 5a, the effect of varying the DV temperature alone while maintaining the same PV conditions is examined. The impact of varying the DV temperature from 18[degrees]C to 20[degrees]C (64.4[degrees]F to 68[degrees]F) on the macroclimate air temperature is clear on all heights, although it is amplified on the lower levels where the DV supply is installed. The agreement between the predicted and measured temperatures was good with an error ranging from -1[degrees]C (1.8[degrees]F) at the supply level to +0.5[degrees]C (+0.9[degrees]F) at the return. The predicted and measured temperatures obtained from varying both DV and PV temperatures are plotted in Figure 5b. Although the difference is clear between cases A3 and A8, the curves in Figure 5b are close to those in Figure 5a. This might evoke the dominating effect of the DV temperature. To further stress this issue, the PV temperatures and flow rates variations with a constant DV temperature of 20[degrees]C (68[degrees]F) are singled out in Figure 5c. A minimal effect of 0.2[degrees]C (0.36[degrees]F) variation on the surrounding air temperature is observed, especially at the layers adjacent to the PV location. This result is expected since the PV flow is assumed to merge with the rising plume and affect its temperature more than the surrounding air temperature. In Figures 5a, 5b, and 5c, the predicted and measured results present a good match, although the model lightly underestimates the lower layers air temperature.
In Figure 5d, the inner centerline plume temperature and the surrounding air temperature are plotted for the heights ranging from above the cylinder till 2.6 m (8.53 ft): the maximum height reached by the plume (corresponding to a nondimensional height [z.sub.1] = 2.8 m [9.19 ft]). The intersection between the surrounding air temperature and the plume temperature corresponds to the height where the buoyancy forces vanish. The model predicted a zero-density-difference at a height of 1.85 m (6.07 ft) while the measured was at 2.18 m (7.15 ft). The model is good in predicting the plume centerline temperatures below this point with a maximum error of +0.6[degrees]C ([+ or -] l.08[degrees]F) and gives lower values above it with an error that could reach -1.1 [degrees]C (-1.83[degrees]F) because it assumes a constant temperature gradient at all heights, while in reality, it is not the case.
[FIGURE 5 OMITTED]
However, since this model has the specificity of predicting the plume air temperatures in conjunction with the personalized ventilation, some measurements for the plume centerline air temperatures above the human body were conducted and the results expressed in terms of the plume-air temperature difference. The plume--air temperature difference is defined as the difference between the centerline plume temperature and the air temperature outside the plume. Figure 6 shows the comparisons between the measured and predicted data. As seen in these figures, the plume-air temperature difference decreases with height until it reaches zero where the buoyancy forces vanish. Above this point, the thermal plume will continue to rise because of the inertial forces. In Figures 6a and 6b, the PV flow rate is maintained at 4 L/s (8.47 [ft.sup.3]/min), and the PV temperature varied between 20[degrees]C and 22[degrees]C (68[degrees]F and 71.6[degrees]F), respectively. At this relatively low flow rate, the results predicted by the model matched those measured, especially at lower heights that are closer to the PV module. However, when the PV flow rate is increased in Figure 6c to reach 7 L/s (14.83 [ft.sup.3]/min) with a temperature of 20[degrees]C (68[degrees]F), the difference between the predicted and measured temperatures was larger above the occupant head (at 1.2 m [3.94 ft] height) with an error of +0.6[degrees]C (+1.08[degrees]F). In Figure 6d, the same flow rate of 7 L/s (14.83 [ft.sup.3]/min) is maintained but with a higher PV temperature of 2T[degrees]C (71.6[degrees]F). In this case, the error is lower at the height of 1.2 m (3.94 ft), and it reaches +0.4[degrees]C (+0.72[degrees]F). This shows that the model accuracy close to the PV module is decreased when the PV flow rate is increased and its temperature decreased. However, in all parts of Figure 6, a maximum error of -l.4[degrees]C (-2.52[degrees]F) is observed at heights above 2.3 m (7.55 ft). This is due to the uniform temperature gradient ([partial derivative][theta]/[partial derivative]z) assumed while calculating [DELTA][[theta].sub.sp,0] in Equation 10, while in fact, this temperature gradient varies at different heights and is much smaller at the upper levels as seen in Figure 5. In addition, some disturbance is observed in Figures 6c and 6d in the plume temperature pattern when the flow rates are increased comparing to the cases in Figures 6a and 6b.
[FIGURE 6 OMITTED]
An improved simple numerical model was developed to represent room heat transfer with DV assisted by personalized ventilation systems. The model is based on separate plume and surrounding air temperatures while accounting for the PV effect on the thermal plumes. The source plumes flow rates and temperatures are based on the work of Mundt (1992) and Morton et al. (1956), while the wall plumes characteristics are obtained from the work of Eckert and Jackson (1950).
The model showed good accuracy in predicting the surrounding air temperature, especially at higher heights for different DV temperatures. However, it underestimates the lower level temperatures by a maximum of-1 [degrees]C (-1.8[degrees]F). The model is able also to predict the thermal plumes temperatures and flow rates at all heights. It has very good accuracy in predicting the plumes temperatures close to the PV modules for moderate PV temperatures and flow rates. However, the accuracy decreases and the model overestimates the centerline plume temperature with the increasing PV flow rate and the decreasing PV temperature.
A = walls area, [m.sup.2] ([ft.sup.2])
[C.sub.p] = specific heat of air, J/kg x K (Btu/lbm-[degrees]F)
dA = elementary cross-sectional area, [m.sup.2] ([ft.sup.2])
d[??] = elementary mass flow rate, kg/s (lbm/h)
g = gravitational acceleration, m/[s.sup.2] (ft/[s.sup.2])
h = specific enthalpy, J/kg (Btu/lbm)
[h.sub.c] = convective coefficient, W/[m.sup.2] x K (Btu/h.[ft.sup.2].[degrees]F)
[??] = enthalpy rate, W (Btu/h)
L = height of the wall, m (ft)
[??] = air mass flow rate, kg/s (lbm/h)
M = air mass flow rate, kg/s (lbm/)
[P.sub.k] = source heat flux, W (Btu/h)
q = plume mass flow rate, kg/s (lbm/h)
Q = heat transfer rate, W (Btu/h)
r = radial distance from plume centerline, m (ft)
R = plume radius, m (ft)
T = temperature, K ([degrees]F)
U = thermal plume velocity, m/s (fl/min)
y = distance from the wall, m (ft)
Y = wall width, m (fl)
z = plume height, m (ft)
[z.sub.v] = plume virtual source point height, m (ft)
[alpha] = entrainment coefficient relating the inflow velocity at the edge of the plume to the vertical velocity within the plume
[beta] = coefficient of cubical expansion for air, [K.sup.-1] ([R.sup.-1])
[delta] = boundary layer thickness, m (fl)
[DELTA][[theta].sub.sp] = source plume-ambient air temperature difference, K ([degrees]F)
[DELTA][[theta].sub.w] = wall-ambient air temperature difference, K ([degrees]F)
[DELTA][[theta].sub.wp] = wall plume-ambient air temperature difference, K ([degrees]F)
[rho] = density of air, kg/[m.sup.3] (lbm/[ft.sup.3])
0 = plume center line
1 = nondimensional form
cir = circulated air
e = exit
i = inlet
s, sp = source plume
sup = displacement ventilation supply air
w, wp = wall plume
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Makhoul Alain, * Ghali Kamel, and Ghaddar Nesreen
Department of Mechanical Engineering, American University of Beirut, Bliss Street, Beirut 1107-2020, Lebanon
* Corresponding author e-mail: ah[m.sup.2]email@example.com
Received March 2, 2011; accepted July 4, 2011
Makhoul Alain is PhD student. Ghali Kamel is Associate Professor. Ghaddar Nesreen, Member ASHRAE, is Professor and Qatar Chair.
Table 1. DV and PV flow rates and temperatures for experimental cases. DV flow rate, DV temperature, Case L/s ([ft.sup.3]/min) [degrees]C ([degrees]F) A1 60 (127) 18 (64.4) A2 60 (127) 18 (64.4) A3 60 (127) 18 (64.4) A4 60 (127) 18 (64.4) A5 60 (127) 20 (68) A6 60 (127) 20 (68) A7 60 (127) 20 (68) A8 60 (127) 20 (68) PV flow rate, PV temperature, Case Us ([ft.sup.3]/min) [degrees]C ([degrees]F) A1 4 (8.47) 20 (68) A2 4 (8.47) 22 (71.6) A3 7 (14.83) 20 (68) A4 7 (14.83) 22 (71.6) A5 4 (8.47) 20 (68) A6 4 (8.47) 22 (71.6) A7 7 (14.83) 20 (68) A8 7 (14.83) 22 (71.6)
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|Author:||Alain, Makhoul; Kamel, Ghali; Nesreen, Ghaddar|
|Publication:||HVAC & R Research|
|Date:||Aug 1, 2012|
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